where \(T>0\)  is the final simulation time, \(u\)  is the unknown solution, such that \(u = (u_1,u_2) : \mathbb {R} \times (0, T) \mapsto \mathbb {R}^n\)  with \(|u| := \sqrt{u_1^2+u_2^2}\) . The initial datum is given by \(u_0 =(u_1^0,u_2^0) \in L^\infty (\mathbb {R}, \mathbb {R}^n)\)  and \(\phi (r)\)  is a scalar function such that \(\phi (r) \in C^1(\mathbb {R}^+)\) , where \(r\phi (r)\rightarrow 0\)  as \(r\rightarrow 0\) . This system
is a prototype of a non strictly hyperbolic system of
conservation laws, serving as a model for the elastic
string (see [1]}). Nevertheless, such model appears in magnetohydrodynamics, where it has been used, for example,
to explain certain features of the solar wind, such as in
[2]}. We can rewrite Eq.
(REF ) in a more explicit form by introducing
variable \(r = |u|\)  to approximate its
strong generalized entropy solution (as in [3]}):
\({\left\lbrace \begin{array}{ll}r_t + (r \phi (r))_x = 0, &\quad (x,t) \in \mathbb {R}\times (0,T), \\u_t + (u \phi (r))_x = 0, &\quad (x,t) \in \mathbb {R}\times (0,T), \\u(x,0) = u_0(x), \; r(x,0) = r_0(x) = |u_0(x)|, &\quad x \in \mathbb {R}.\end{array}\right.}\) 
